Hydrothermal operation is a well-known large-scale, real-life, application of stochastic multistage optimization techniques. Algorithms such as hybrid Dynamic Programming and Stochastic Dual Dynamic Programming (SDDP/DP) have been successfully applied to these problems, where SDDP with weekly stages is used to manage inflow uncertainty, usually represented as an autoregressive stochastic model. In systems with a high penetration of renewables, there is another level of uncertainty at the intra-week scale: the variability of energy production from wind and solar sources, combined with net load variability driven by factors such as temperature and cloud cover (e.g. due to air conditioning and rooftop solar).
The objective function of this week-ahead scheduling problem (WASP) is to minimize the expected value of the sum of operation costs along the hours (or 15-minute intervals) along the week plus the expected future operation cost at the end of the week, represented by the well-known future cost function produced by the SDDP/DP algorithms.
In the authors’ experience, the main challenges in solving WASP are: (i) the linear stochastic models used in SDDP/DP are not adequate to represent the complex, nonlinear relationship between wind, solar and net load; (ii) the effect of diverse forecasting techniques for these value has to be considered in the optimization process; (iii) it is also necessary to represent time-coupling constraints such as ramps on hydro outflow and forebay variation, minimum uptime and downtime for the committed thermal plants.
This paper presents a hybrid probabilistic solution approach to WASP based on the following methodologies and assumptions: (i) Integrated scenarios of hourly renewable production and net loads, plus weekly/daily inflows to the hydro plants; (ii) A probabilistic forecasting model assigns weights to these scenarios, reflecting the fact that forecasting accuracy varies with time; (iii) A spatial/severity clustering algorithm to divide the scenarios into subsets; and (iv) A multivariate set of affine functions, related to the clusters identified in step (iii), is used as part of a multistage MIP optimization problem to determine the optimal scheduling under uncertainty for the weighted scenarios produced in steps (i) and (ii).
The above approach will be illustrated with the scheduling of the detailed generation-transmission system of Chile, which has a complex mix of hydro, wind, solar and fossil fuel plants such as combined cycle gas and coal.